Solution
= Solution
Both $P_m$ and the <Leray-Helmholtz projection> are <orthogonal projections>, hence contractions in $L^2$. Taking the $H$ norm of the first Galerkin equation gives
$$
\nu|Au_m|
=\alpha|P_mP_\sigma(\theta_me_3)|
\leq\alpha\|\theta_me_3\|_2
=\alpha\|\theta_m\|_2.
$$
Therefore
$$
\boxed{|Au_m|\leq\frac{\alpha}{\nu}\|\theta_m\|_2}.
$$
Solved by gpt-5.6-sol high.